Circular Functions (Trigonometry)

Circular functions Revision

Where do cos,sin and tan come from?

Unit circle (of radius 1)

cos is the x – coordinate

sin is the y – coordinate

cos

sintan

all 3 are measures of length.

Remember SOH CAH TOA

Exact values:

0

30o

6

45 o

4

60 o

3

90 o

2

180 o

270 o

2

3

360 o

2

cos 1 2

3

2

2

2

1

2

1 0 -1 0 1

sin 0 2

1

2

2

2

1

2

3 1 0 -1 0

tan 0 3

3

3

1 1 3 undefined 0 undefined 0

Angle conversions (between radians and degrees).

Quadrants and symmetry:

o All Students Talk C.. (ASTC)

Finding Exact values:

Example: What is the exact value of:

(a) 4

5sin

; (b)

3

2tan

.

(a) 1. Sign: 3rd Quadrant -ve

2. Angle Equivalent (1st Quadrant): 44

5

4

3. So: 2

2

2

1

4sin

4

5sin

or

(b) 1. Sign: 2nd “negative” Quadrant +ve

2. Angle Equivalent (1st Quadrant): 33

2

3

3. So: 33

tan3

2tan

Jump Start Holiday Questions

Review: radians, definitions, exact

values, symmetry

Ex6A Q 1, 2, 3, 4 (ace for all);

Ex6B Q 1, 2acegik, 3 acegikmoqsu, 4 aceg,

5 abdfgj, 6

Ex6C Q 2

*CALCULATOR MODE: Always work in radians*

F X D r a w

AS

T C

Solving equations involving circular functions.

Finding axis intercepts:

1. Y-intercepts:

)0(f or 0x .

E.g. what is the Y-intercept of 26

2sin3)(

xxf

o

2

4332

2

33)0(

22

33)0(

23

sin3)0(

26

2sin3)0(

26

02sin3)0(

f

f

f

f

f

2. X-intercepts:

0)( xf or 0y .

Examples: Find all values of for:

(a)

2,0,2

3cos:

1. Sign: (+ve) Q1, 4

2. Angle: 6

3.

6

11,

6

62,

6

(b) 2,0,7.0sin:

1. Sign: (-ve) Q3, 4

2. Angle: 7754.0)7.0(sin 1

3. 5078.5,9170.3

7754.02,7754.0

(c) 2,2,01sin2:

Rewrite: 2

1sin

1. Sign: (-ve) Q3, 4, -1, -2

2. Angle: 6

3.

6

11,

6

7,

6,

6

5

62,

6,

60,

6

(d) 2,0,022cos4

Rewrite: 2

1cos

Let 4,02 aa

2

1cos a

1. Sign: (-ve) Q 2, 3, 6, 7

2. Angle: 3

3.

3

5,

3

4,

3

2,

3

3

10,

3

8,

3

4,

3

22

33,

33,

3,

3

a

a

Ex6E 1 ace, 2 ac, 3 ac, 4 ab, 5 abc, 6 ace, 7 ace, 8 acegi; Ex6J 4, 5, 6

2011 Exam1 Question

Graphs of Circular Functions siny cosy

x-2 – 2

y

-2

-1

1

2

x-2 – 2

y

-2

-1

1

2

Period = 2

Amplitude = 1

Range: [-1, 1]

tany

x-2 – 2

y

-10

-5

5

10

Period =

We don’t refer to the amplitude for tany

Range: R

nperiod

2

nperiod

Transformations of siny & cosy

siny cbnay sin & cosy cbnay cos

a: a dilation of factor “a” from the x-axis.

n: a dilation of factor “n

1” from the y-axis.

b: a translation of b units along the x-axis.

c: a translation of c units along the y-axis.

1. Dilations

(a) The effect of “a”

Graph the following graphs: (i) cos2y ; (ii) 2

siny ; where 2,0

(i) (ii)

“a” affects the amplitude.

(b) The effect of “n”

Graph the following graphs: (i) 2cos3y ; (ii)

2sin3

y ; where 2,0

(i) (ii)

“n” affects the period.

n

period2

2. Reflections.

Two types:

o Reflection in the x-axis: )(xf

o Reflection in the y-axis: )( xf

Examples:

Sketch the graphs of the following:

(a) 2sin3y ; (b)

3cos2

y ; where 2,0

(a) (b)

3. Translations

(a) The effect of “c”

Sketch the following: (i) 3sin3 y ; (ii) 32cos2 y ; where 2,0

(i) (ii)

x2

32

2

y

-3

-2

-1

1

2

3

x1 2 3 4 5 6

y

-3

-2

-1

1

2

3

x4

2

34

54

32

74

2

y

-6-5-4-3-2-1

123

(b) The effect of “b”

Sketch the following:

(i)

4sin2

y ; (ii)

32cos3

y ; where 2,0

(i) (ii)

Combining all transformations

Example: Sketch the graph of ]2,0[,22

2sin3)(

f

Rewrite: 24

2sin3)(

f

22,4

,3 nandcba

Sketch 2sin3)( f first: Secondly with translations:

Note: X-intercepts need to be found!!

Ex6F 1 adfhi, 2, 4, 5; Ex6G 1, 2 ac, 3 ef, 5 acfgh, 6, 7

x4

2

34

54

32

74

2

y

-3-2-1

123456

x4

2

34

54

32

74

2

y

-3-2-1

123456

x4

2

34

54

32

74

2

y

-6-5-4-3-2-1

123456

x4

2

34

54

32

74

2

y

-6-5-4-3-2-1

123456

Graphs &Transformations of the Tangent function

Example: Sketch 6

13

632tan3

xforxy

Rewrite:

62tan3

xy

x2

32

2 52

3

y

-10-8-6-4-2

2468

10

Ex6J 1, 2, 7, 8, 9

Addition of ordinates (add the ‘y’ values)

Example:

(a) On the same set of axes sketch xxf sin2)( and xxg 2cos3)( for

20 x ;

(b) Use addition of ordinates to sketch the graph of xxy 2cos3sin2 .

Note: For )2cos(3)sin(2 xxy it is easier to do ))2cos(3()sin(2 xxy

Ex6H 1 ace

Solving Equations where both sin & cos appear

Example: Solve for 2,0, xx :

(i)

5.0tan

cos

cos5.0

cos

sin

cos

cos5.0sin

x

x

x

x

x

xbysidesbothdivide

xx

1. Sign: (+ve) Q 1, 3

2. Angle: 464.0)5.0(tan 1

3. 605.3,464.0

464.0,464.0

x

x

(ii) 03cos33sin xx

9

16,

9

13,

9

10,

9

7,

9

4,

9

3

16,

3

13,

3

10,

3

7,

3

4,

33

33tan

033tan

03cos33sin

x

x

x

x

xx

Ex6J 10, 11 acegi, 12

General Solutions to Circular Functions

Example: Solve 2

1cos x

Solution:

ZnnnnChecknx

generally

x

x

Angle

Cos

x

1,1,0:3

2

:

.....,3

11,

3

7,

3

5,

33,

3

5.....,

,...3

4,3

2,3

2,3

,3

,3

2.....3

3:.2

.... 9, 5, 4, 1, 4,-1,-... Quad positive .1

2

1cos

or

3

65,

3

61

23

5,2

3

nn

nn

So in general terms: Znanxax

),(cos2cos

1

Example: Solve 2

1sin x

Solution:

1,1,0:6

52

6)12(

1,1,0:6

2

:

.....,6

17,

6

13,

6

5,

66

7,

6

11.....,

,...6

3,6

2,6

,6

,6

,6

2.....3

6:.2

.... 6, 5, 2, 1, 4,-3,-... Quad positive .1

2

1sin

nnnChecknnx

or

ZnnnnChecknx

generally

x

x

Angle

Sin

x

So in general terms: Znanoranx

ax

),(sin)12()(sin2

sin11

The above can be simplified to Znanx n ,)(sin)1( 1

For Znanx

ax

,)(tan

tan1

Example 1: Find the general solution for 13

sin2

x

Solution:

22

2

3)12(

6

52

36

7)12(

36

72

6

7)12(

36

72

3

2

1sin)12(

32

1sin2

3

2

1

3sin

13

sin2

)(sin)12()(sin2

11

11

nnxornx

nxornx

nxornx

nxornx

x

x

anxoranx

Example 2: Find the general solution to 24

2cos2

x , and hence find all the solutions from

2,2 .

Solution:

4

7,,

4

3,0,

4,,

4

54

72

4

31

400

4

54

9

4

22222

42

42

2

2cos2

42

2

2

42cos

24

2cos2

)(cos2

1

1

x

4-2x & domain) in (not 2x n

4-x & x n

4-x & x n

4--x & -x -1n

domain) in (not 4

--2x & domain) in (not -2x -2n

2 ,2- domain of solutionsfor

solutiongeneralnxornx

nxornx

nx

nx

x

x

anx

Ex6K 1, 2, 3, 6ab, 8, 9

Determining Rules for Circular Functions

Example: The graph shown has the rule of the form: cbtnay )(cos , find ncba &,, .

x–

3–

66

3

y

-3

-2

-1

1

21,3: arangea

11: cyrangeofmiddlec

32

3

22: n

nnperiodn

b

b

b

b

b

curvetheonisor

bknowledgepreviousfromb

3

3

3cos1

3cos22

1)0(3cos23

:3,0

3::

Ex6I 1, 2, 3, 4, 5, 6, 7, 8, 9; Ex6J 14, 15

Applications of Circular Functions

Rewrite:

512

)4(cos7

tT

2

35

2

75

3cos75

12

)40(cos7,0

24

12

2

12,5,4,7

Ttat

hoursperiod

ncba

The machine will not be able to operate for 12 hours, i.e 10 ≤ t ≤ 22. (i.e from 10a.m.

to 10 p.m.)

Ex6L 1, 2, 4, 6 Ex 6N

Past Exam Questions

2008

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2011

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2014

2015